Complex Surface Geometry in Nano-Structure Solids: Fractal versus Bernal-Type Models

  • Peter Pfeifer
  • David Avnir
  • Dina Farin
Part of the NATO ASI Series book series (NSSB, volume 258)


In this chapter we present a discussion of a controversy that has recently arisen and that has been witnessed at this conference in the lectures by J. Klafter and by one of us (P.P.). The controversy concerns the question whether the fractal concept, as found to be highly successful in a large number of surface science problems (see Refs. 1-3 for recent surveys), is applicable to a certain class of porous silicas as reported by us and others,4-11 or whether these silicas are more appropriately described by traditional random-packed sphere models as proposed by Drake, Levitz, and Klafter (DLK).12-16 The fractal model asserts that the surface geometry scales with dimension D ≈ 3 and thus is highly disordered, from atomic length scales upward. The random-packed sphere model, to which we shall refer as Bernal-type model because of its similarity to the familiar Bernal model for liquids,17-19 describes the solid as a random assembly, more or less close-packed, of hard spheres with fixed diameter. A typical value of the sphere diameter, as proposed by DLK, is 70 Å. Their model thus asserts that the surface is smooth and scales with dimension D = 2 from atomic lengths up to the sphere diameter. The question then, which of the two models describes the structure of the silicas more adequately, is a rather specialized one and may seem of limited interest. But DLK have elevated it to an issue from which they wish to conclude that “the situation with surface fractals is less clear and still controversial,”15 as if to build a general case against the fractal concept in surface science. We therefore believe it is important to clarify the issue, to assess the two models in an unbiased way, and to obtain, from the analysis, guidelines for similar studies of other systems.


Fractal Model Malachite Green Sphere Model Packing Fraction Sphere Diameter 
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Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Peter Pfeifer
    • 1
  • David Avnir
    • 2
  • Dina Farin
    • 2
  1. 1.Department of Physics and AstronomyUniversity of MissouriColumbiaUSA
  2. 2.Department of Organic Chemistry and F. Haber Research Center for Molecular DynamicsThe Hebrew University of JerusalemJerusalemIsrael

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